Constructing Driver Hamiltonians for Optimization Problems with Linear Constraints

TL;DR

This paper introduces an algebraic framework for constructing driver Hamiltonians that commute with linear constraints in quantum optimization, revealing the problem's NP-Complete complexity and impacting quantum algorithms like QAOA.

Contribution

It provides a simple, intuitive algebraic method to analyze and classify the complexity of designing driver Hamiltonians for arbitrary linear constraints.

Findings

Classifies the problem of finding commuting driver Hamiltonians as NP-Complete.

Provides an algebraic framework applicable to quantum annealing and QAOA.

Enables more efficient embedding of problems on quantum devices by using specialized Hamiltonians.

Abstract

Recent advances in the field of adiabatic quantum computing and the closely related field of quantum annealers has centered around using more advanced and novel Hamiltonian representations to solve optimization problems. One of these advances has centered around the development of driver Hamiltonians that commute with the constraints of an optimization problem - allowing for another avenue to satisfying those constraints instead of imposing penalty terms for each of them. In particular, the approach is able to use sparser connectivity to embed several practical problems on quantum devices than other common practices. However, designing the driver Hamiltonians that successfully commute with several constraints has largely been based on strong intuition for specific problems and with no simple general algorithm to generate them for arbitrary constraints. In this work, we develop a simple and intuitive algebraic framework for reasoning about the commutation of Hamiltonians with linear constraints - one that allows us to classify the complexity of finding a driver Hamiltonian for an arbitrary set of constraints as NP-Complete. Because unitary operators are exponentials of Hermitian operators, these results can also be applied to the construction of mixers in the Quantum Alternating Operator Ansatz (QAOA) framework.